Verify that $f(x)=\frac{1}{x}-\frac{1}{x_0}$ is continuous for every $x_0\neq 0$.

Question

Verify that $f(x)=\frac{1}{x}-\frac{1}{x_0}$ is continuous for every $x_0\neq 0$.

$f(0)$ is not defined. So the function is discontinuous at $0$.

Let $c\in \mathbb{R}\setminus \lbrace 0 \rbrace$, we have that

$\lim_{x \to c} f(x)=\lim_{x \to c} (\frac{1}{x}-\frac{1}{x_0})= \frac{1}{c}-\frac{1}{x_0}=f(c)$

Then $f$ continuous on $\mathbb{R}\setminus \lbrace 0 \rbrace $.

Is that true, please?